Illumination device for projection system and method for fabricating

ABSTRACT

A masking aperture for a photomask illumination system provides controlled on-axis and off-axis illumination. The masking aperture has a dithered pattern of pixels. The intensity of the pattern controls the illumination of the photomask. The masking aperture pattern defines one or more zones of illumination. Zones comprise elements that are patterned in accordance with a selected wavelength of incident light to diffract the incident light into an illumination pattern for illuminating a photomask. Each of the elements is constructed with a matrix of pixels. In the preferred embodiment the array of pixels is 8×8. The number of elements is generally greater than 3×3.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is a division of U.S. patent application Ser.No. 09/422,398, filed Oct. 21, 1999, now U.S. Pat. No. 6,466,304 whichclaims the benefit of the priority date of the following U.S.Provisional Application No. 60/105,281 filed Oct. 22, 1998 and U.S.Provisional Application No. 60/119,780, filed Feb. 11, 1999, each ofwhich applications are hereby incorporated in their entirety into thepresent application by reference.

Reference is also made to co-pending application entitled “IlluminationDevice for Projection System and Method for Fabricating,” being filedconcurrently herewith and hereby incorporated in its entirety into thepresent application by reference.

BACKGROUND OF THE INVENTION

Optical lithography has been one of the principal driving forces behindthe continual improvements in the size and performance of the integratedcircuit (IC) since its inception. Feature resolution down to 0.30 μm isnow routine using the 365 nm mercury (Hg) i-line wavelength and opticalprojection tools operating at numerical apertures above 0.55 withaberration levels below 0.05 λ RMS OPD. The industry is at a point whereresolution is limited for current optical lithographic technologies. Inorder to extend capabilities toward sub-0.25 μm, modifications in sourcewavelength, optics, illumination, masking, and process technology arerequired and are getting very much attention.

However, as devices get smaller, the photomask pattern becomes finer.Fine patterns diffract light and thus detract from imaging the photomaskonto the surface of a wafer. FIG. 1a shows what happens when a photomaskwith a fine pattern 6 having a high frequency (pitch 2 d is aboutseveral microns), is illuminated through a projection lens system 7. Thefine pattern 6 is illuminated along a direction perpendicular to thesurface thereof and it diffracts the light that passes through the mask6. Diffraction rays 3-5 caused by the pattern include a zero-th orderdiffraction ray 4 directed in the same direction as the direction ofadvancement of the input ray, and higher order diffraction rays such aspositive and negative first order diffraction rays 3, 5, for example,directed in directions different from the input ray. Among thesediffraction rays, those of particular diffraction orders such as, forexample, the zero-th order diffraction ray and positive and negativefirst order diffraction rays 3, 5, are incident on a pupil 1 of theprojection lens system 7. Then, after passing through the pupil 1, theserays are directed to an image plane of the projection lens system,whereby an image of the fine pattern 6 is formed on the image plane. Inthis type of image formation, the ray components which are contributableto the contrast of the image are higher order diffraction rays. If thefrequency of a fine pattern increases, it raises a problem that anoptical system does not receive higher order diffraction rays.Therefore, the contrast of the image degrades and, ultimately, theimaging itself becomes unattainable.

As will be shown below, some solutions to this problem rely upon shapingthe rays of light impinging the photomask in order to provide off-axisillumination to compensate for the lost contrast due to diffraction.These techniques rely upon optical systems for shaping the rays thatilluminate the photomask.

In considering potential strategies for sub-0.25 μm lithography, theidentification of purely optical issues is difficult. Historically, theRayleigh criteria for resolution (R) and depth of focus (DOF) has beenutilized to evaluate the performance of a given technology:

R=k ₁ λ/NA

DOF=+/−k ₂ λ/NA ²

where k₁ and k₂ are process dependent factors, λ is wavelength, and NAis numerical aperture. As wavelength is decreased and numerical apertureis increased, resolution capability improves. Considered along with thewavelength-linear and NA-quadratic loss in focal depth, reasonableestimates can be made for system performance. Innovations in lithographysystems, materials and processes that are capable of producingimprovements in resolution, focal depth, field size, and processperformance are those that are considered most practical.

The Hg lamp is a source well suited for photolithography and is reliedon almost entirely for production of radiation in the 350-450 nm range.Excimer lasers using argon fluoride (ArF) and krypton fluoride (KrF),which produce radiation at 193 nm and 248 nm, respectively, are alsoused. DUV lithography at 248 nm is now being implemented intomanufacturing operations and may be capable of resolution to 0.18 μm

The control of the relative size of the illumination system numericalaperture has historically been used to optimize the performance of alithographic projection tool. Control of this NA with respect to theprojection systems objective lens NA allows for modification of spatialcoherence at the mask plane, commonly referred to partial coherence.This is accomplished through specification of the condenser lens pupilsize with respect to the projection lens pupil in a Köhler illuminationsystem. Essentially, this allows for manipulation of the opticalprocessing of diffraction information. Optimization of the partialcoherence of a projection imaging system is conventionally accomplishedusing full circular illuminator apertures. By controlling thedistribution of diffraction information in the objective lens with theilluminator pupil size, maximum image modulation can be obtained.Illumination systems can be further refined by considering variations tofull circular illumination apertures. A system where illumination isobliquely incident on the mask at an angle so that the zero-th and firstdiffraction orders are distributed on alternative sides of the opticalaxis may allow for improvements. Such an approach is generally referredto as off-axis illumination. The resulting two diffraction orders can besufficient for imaging. The minimum pitch resolution possible for thisoblique condition of partially coherent illumination is 0.5 λ/NA, onehalf that possible for conventional illumination. This is accomplishedby limiting illumination to two narrow beams, distributed at selectedangles. The illumination angle is chosen uniquely for a given wavelength(λ), numerical aperture (NA), and feature pitch (d) and can becalculated for dense features as sin⁻¹ (0.5 λ/d) for NA=0.5 λ/d. Themost significant impact of off axis illumination is realized whenconsidering focal depth. In this case, the zero-th and 1st diffractionorders travel an identical path length regardless of the defocus amount.The consequence is a depth of focus that is effectively infinite.

In practice, limiting illumination to allow for one narrow beam or pairof beams leads to zero intensity. Also, imaging is limited to featuresoriented along one direction in an x-y plane. To overcome this, anannular or ring aperture has been employed that delivers illumination atangles needed with a finite ring width to allow for some finiteintensity. The resulting focal depth is less than that for the idealcase, but improvement over a full circular aperture can be achieved. Formost integrated circuit application, features are limited to horizontaland vertical orientation, and a four-zone configuration may be moresuitable. Here, zones are at diagonal positions oriented 45 degrees tohorizontal and vertical mask features. Each beam is off-axis to all maskfeatures, and minimal image degradation exists. Either the annular orthe four-zone off-axis system can be optimized for a specific featuresize, which would provide non-optimal illumination for all others. Forfeatures other than those that are targeted and optimized for, higherfrequency components do not overlap, and additional spatial frequencyartifacts are introduced. This can lead to a possible degradation ofimaging performance.

When considering dense features (1:1 to 1:3 line to space duty ratio),modulation and focal depth improvement can be realized through properchoice of illumination configuration and angle. For true isolatedfeatures, however, discrete diffraction orders would not exist; insteada continuous diffraction pattern is produced. Convolving such afrequency representation with either illumination zones or annular ringswould result in diffraction information distributed over a range ofangles. Truly isolated line performance is, therefore, not improved withoff-axis illumination. When features are not completely isolated buthave low density (>1:3 line to space duty ratio), the condition foroptimum illumination will not be optimal for more dense features.Furthermore, the use of off-axis illumination is generally not requiredfor the large pitch values that correspond to low density geometry. Asdense and mostly isolated features are considered together in a field,it follows that the impact of off-axis illumination on these featureswill differ, and a large disparity in dense to isolated featureperformance can result.

One approach to generate off-axis illumination is to incorporate a metalaperture plate filter into the fly eye lens assembly of the projectionsystem illuminator providing oblique illumination. A pattern on such ametal plate would have four quadruple openings (zones) with sizing andspacing set to allow diffraction order overlap for specific geometrysizing and duty ratio on the photomask, as disclosed in JP patentLaid-Open (KOKAI) Publication No. 4-267515. Such an approach results ina significant loss in intensity available to the mask, loweringthroughput and making the approach less than desirable. Additionally,the four circular openings need to be designed specifically for certainmask geometry and pitch and would not improve the performance of othergeometry sizes and spacings. Large levels of mask biasing or maskoptical proximity correction (OPC), where mask features arepre-distorted to produce desired image characteristics, would berequired to allow for use of this approach with a variety of features.Filtering, by limiting its effective area, reduces the effect of the flyeye diffuser on maximizing illumination uniformity. Illuminationuniformity may be degraded. This approach also limits the illuminationprofile to one having holes in a metal plate. That is, the masking metalmust remain contiguous. The previous work in this area describes suchmethods using either two or four openings in the aperture plate:EP0500393, U.S. Pat. No. 5,305,054, U.S. Pat. No. 5,673,103, U.S. Pat.No. 5,638,211, EP0496891, EP0486316, U.S. Pat. No. 37,9252.

Another approach to off-axis illumination using the four-zoneconfiguration, which is disclosed in U.S. Pat. No. 5,627,625, is todivide the illumination field of the projection system into beams thatcan be shaped to distribute off-axis illumination to the photomask. Byincorporating the ability to shape off-axis illumination, throughput andflexibility of the exposure source is maintained. Additionally, thisapproach allows for illumination that combines off-axis and on-axis(conventional) characteristics. By doing so, the improvement to densefeatures that are targeted with off-axis illumination is lesssignificant than straight off-axis illumination. The performance of lessdense features, however, is more optimal because of the more preferredon-axis illumination for these features. The result is a reduction inthe optical proximity effect between dense and isolated features.Optimization is less dependent on feature geometry and more universalillumination conditions can be selected.

A problem with this divided illumination approach is that it requiresreconfiguration of the illumination system of a projection tool, a taskthat is not practical on existing tools or systems designed with otherillumination systems. Additionally, the use of divided beam illuminationlimits the fine control of beam shape, size, and position to that whichis possible with optical components utilized in the system. Variationsin shape, size, position, number of beams, maximum aperture size, orother feature or lens specific variations to the illumination intensityprofile become difficult without significant mechanical modifications.Some variations may not be practical or possible with this approach.This has significantly limited the acceptance or use of this approach inmost integrated circuit fabrication operations.

SUMMARY OF THE INVENTION

A shaped illumination approach is described that allows for off-axisillumination of a photomask in a projection exposure tool. It isnecessary to control both the off-axis and the on-axis character of theillumination for instance so that dense line performance can be improvedand more isolated line performance can be maintained, i.e., opticalproximity effect (hereinafter “OPE”) is kept to a minimum. Minimal OPEcorrection is desired to reduce mask complexity. There is also a desirefor a flexible technique that can be incorporated into most existing orfuture generation projection exposure tools with a minimum amount ofillumination system retrofitting. It is important that such animprovement be easily removed to allow a return to original operationconditions since it is expected that a given projection exposure systemwould be used for a variety of applications. Such an approach can leadto optimizing illumination conditions, which can be incorporated into anexposure system as a more permanent condition. The invention alsoprovides an improvement that allows for fine adjustment or modificationto accommodate mask-specific or lens-specific characteristics isimportant as resolution and focal depth requirement are pushed beyondthe capability of conventional optical lithography. Maintainingillumination throughput is also critical, as any loss in intensity willresult in increased exposure time requirements.

The invention provides such a solution. It modifies existingillumination system by adding a masking aperture in the illuminationpupil plane, fabricated as an optical component reticle, patterned anddithered into a large number of elements to allow for control of theprojected light distribution at the mask plane and inserted at thecondenser lens pupil plane. This masking aperture comprises atranslucent substrate and a masking film. The distribution of theintensity through the masking aperture in the illumination pupil plane,is determined to provide optimized illumination. The illumination regionor regions exhibit varying intensity, which is accomplished by creatinga half-tone pattern via pixelation of the masking film, thereby allowingfor maximum variation in illumination beyond the simple binary (clear oropaque) possible with earlier pupil plane filtering approaches.

More specifically, the invention includes an aperture mask for anillumination system to provide controlled on-axis and off-axisillumination. The aperture mask acts as a diffraction element. Theaperture mask is divided into an array of elements and each elementcontains an array of pixels. Each of the elements is constructed with amatrix of pixels. In the preferred embodiment the array of pixels is8×8. The number of elements in the illumination array are generallylarger than 3×3 and array sizes of 21×21 and 51×51 are an example, whichcorrespond to 441 and 2601 elements respectively. The elements arepatterned in accordance with a selected wavelength of incident light todiffract the incident light into an illumination pattern forilluminating a photomask.

The intensity is modulated by the intensity state of pixels within eachelement. The highest intensity element has all pixels clear or maximumintensity. Light of suitable wavelength passes through withoutattenuation. An element with 64 pixels having dark or minimum intensityattenuates or blocks all light. Elements of intensity between none (0%)and all (100%) are created by the state of the pixels in a givenelement. Random patterns and other patterns between elements may produceartifacts similar to moire patterns. Such artifacts are undesired. Idiscovered that a dithered pattern using position dependent thresholdsproduced illumination patterns that had little or no artifacts.

I also discovered that traditional, optical shaping systems such asbeam-splitters or diffractive optical elements and my diffractionshaping system can each be improved by adding an illumination aperture.A square illumination aperture shows maximum improvement. It is locatedat or near the pupil of the illumination system. An aperture with alarge, central square opening and an opaque border is inserted at thecondenser lens pupil proximate to the fly's eye lens. It can also bedesigned into the masking aperture. The resulting illumination patternfills the corners and squares the edges of the illumination pupil andlimits the non-optimal frequency spreading character along the x and yaxes while optimizing the off-axis illumination angles. The squareillumination aperture is especially helpful for imaging features thatare oriented along x and y directions in the mask plane. The use of acentral obscuration (square and also round shaped) applied in the samelocation can similarly be achieved and can lead to performanceimprovements described hereinafter. Furthermore, any combination ofoff-axis illumination with a square pupil or obscuration has potentialto improve performance for geometry oriented in the x-y direction. Thiscan include, but is not limited to, round zones, elliptical zones,square zones, and annular slots (that is an annular ring masked off on xand y axis to form arc-shaped zones).

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1a is a schematic view of an illumination system for a photomask.

FIG. 1b shows the zero-th and first positive and first negativediffraction rays.

FIG. 1c is a schematic view of a four-zone illumination arrangement.

FIG. 2 is a drawing showing diffraction orders (first and zero) for 130nm x-oriented features using four-zone illumination, plotted in thefrequency plane of the objective lens. Center sigma is 0.68 and radiussigma is 0.13, chosen to accommodate feature pitch values from 260 to455 nm.

FIG. 3 is a drawing of the continuous tone intensity distribution for amasking aperture based on four distributed-intensity zones.

FIG. 4 is a plot of the x-y distribution of dithered bilevel maskingcells for an illumination aperture consisting of four circular normaldistributed-intensity zones placed at diagonal positions correspondingto off-axis illumination for geometry oriented in horizontal andvertical directions. The axes are divided into relative distances usingthe center and the edges of the mask.

FIG. 5 is a three dimensional plot of the x-y distribution shown in FIG.4.

FIG. 6 is a drawing of surface contours of the continuous tone intensitydistribution for a masking aperture based on two distributed-intensityzones where the maximum off-axis angle is limited to one direction.

FIG. 7 is a plot of the x-y distribution of dithered, bilevel maskingcells for an illumination aperture consisting of four elliptical normaldistributed-intensity zones placed at diagonal positions correspondingto off-axis illumination for geometry oriented in horizontal andvertical directions. The axes are divided into relative distances usingthe center and the edges of the mask.

FIG. 8 is a contour plot of the x-y distribution shown in FIG. 7.

FIG. 9 is a three dimensional plot of the x-y distribution for anillumination aperture consisting of four stepped squaredistributed-intensity zones placed at diagonal positions correspondingto off-axis illumination for geometry oriented in horizontal andvertical directions.

FIG. 10 is a contour plot of the x-y distribution for an illuminationaperture consisting of four stepped square distributed-intensity zones,as described for FIG. 9.

FIG. 11 is a plot of the x-y distribution of dithered, bilevel maskingcells for an illumination aperture consisting of four elliptical normaldistributed-intensity zones, as described for FIG. 9.

FIG. 12 is a schematic representation of the bilevel representation of65 gray levels using the ordered dithering algorithm.

FIG. 13 is a graph showing normalized aerial image log-slope (NILS) vsfocus for 130 nm features using four-zone illumination, δ_(c)=0.68 andδ_(r)=0.20.

FIG. 14 is a graph showing normalized aerial image log-slope (NILS) vs.focus for 130 nm features using distributed-intensity four-zoneillumination δ_(c)=0.68 and δ_(r)=0.30) and a circular hard stop(δ=0.8).

FIG. 15a is a graph showing normalized image log slope (NILS) vs. focusfor 130 nm features using the illumination shown in FIG. 19.

FIG. 15b is a graph showing normalized image log slope (NILS) vs. focusfor 130 nm features using the illumination shown in FIG. 23.

FIG. 15c is a graph showing normalized image log slope (NILS) vs. focusfor annular ring illumination of 150 nm features using 248 nm wavelengthand 0.63 NA, with duty ration of 1:1.5 to 1:5.

FIG. 15d is a graph showing normalized image log slope (NILS) vs. focusfor square ring illumination of 150 nm features using 248 nm wavelengthand 0.63 NA, with duty ration of 1:1.5 to 1:5.

FIG. 16 is a plot of the normalized image log slope, showing influenceof a circular four-zone δ_(c)=0.7 and δ_(r)=0.2 on a zero intensityfield) with 0.1 waves of primary aberrations: astigmatism, tilt,spherical, and coma, through focus for a 130 nm features imaged with a193 nm imaging system using a numerical aperture of 0.6.

FIG. 17 is a plot of the normalized image log slope, using the bi-levelrepresentation of the distributed-intensity illumination described forthis invention.

FIG. 18 is a contour plot of circular gaussian zones used to optimizeenergy distribution in the pupil, where energy is concentrated in thecenter of the pupil and at off-axis angles.

FIG. 19 is a three dimensional plot of circular gaussian zones and asquare limiting zone used to limit the on-axis illumination componentwhile retaining off-axis zone energy.

FIG. 20 is a perspective view of a beamsplitter illumination system witha square aperture mask inserted in its pupil.

FIG. 21 shows how a square central obscuration can be included in anaperture to reduce the on-axis character of illumination and improve theperformance of dense features.

FIG. 22 shows the square ring, off-axis source which delivers optimaloff-axis illumination for dense features while providing on-axisillumination for more isolated features.

FIG. 23 shows how the square ring approach can be combined with thegaussian four-zone approach to emphasize corner zone effects.

FIG. 24 is a contour plot of a combined annular and gaussian four-zoneaperture to improve the off-axis character of annular illumination.

FIG. 25 is an illumination system using diffractive optical elements anda square aperture.

FIG. 26 is a 51×51 element array of a dithered half tone pattern.

DETAILED DESCRIPTION OF THE INVENTION

For given ranges of feature types and/or sizes (types being lines,spaces, contacts, and dense or isolated combinations of these) exposureconditions are optimized by determining the average off-axis angle toaccommodate all features. The distribution of off-axis angles is thendetermined based on the range of feature sizes of interest. As the rangeof feature sizes increases, the condition of off-axis illuminationapproaches a limit equivalent to the on-axis condition. Most commonly,duty ratios for a given feature size may range from 1:1 to 1:6 line:space ratio. The spread of off-axis illumination angles is accomplishedby shaping zones (for the two or more zone, including four-zone) orrings (for the annular) to produce continuous intensity distributions.

As an example of the design considerations, 130 nm features areconsidered using a 193 nm exposure wavelength and a 0.60 objective lensnumerical aperture. Duty ratios from 1:1 to 1:6 are included. Featuresare considered dense with a duty ratio less than 1:3. Table 1 shows thepitch values (p) for these dense features, along with the required axiscenter sigma and four-zone center sigma (δ_(c)) values for optimumoff-axis illumination. Center sigma values on axis are determined asλ/(2p•NA), as shown in FIG. 1. Since diffraction order placement isdetermined by the projection of the four-zone onto the x or y axis, thecenter four-zone sigma values are larger by a (2)^(1/2) factor.

In order to design an off-axis illumination configuration that canaccommodate and enhance the range of pitch values in Table 1, zoneposition and radius values must be chosen so that some amount of orderoverlap occurs for each case. This can be accomplished if the four-zonecenter δ_(c) and the radius δ_(r) values are set as follows:

δ_(c) =SQRT(2)*(0.61+0.35)/2=0.68

δ_(r)>(0.61−0.35)/2=0.13

These choices for zone center and radius values correspond to thesituation where zero and first diffraction orders begin to overlap forthe extreme pitch values of 260 and 455 nm, as shown in FIG. 2. The(2)^(1/2) term does not factor into determination of δ_(r) values sincethe zone radius is projected directly onto the axes. The extent of orderoverlap for these extreme pitch values is, however, mostly ineffectual,and larger δ_(r) values are required to influence performance. A δ_(r)value of 0.20 allows for significant order overlap (˜20%) and is a morepractical starting value for further optimization.

When imaging of both dense and isolated features, illumination thatresembles both the strong four-zone and the conventional on-axisillumination is desirable. This can lead, for instance, to the four-zoneillumination design where circular zones are replaced with continuoustone zones, as shown in FIG. 3.

TABLE 1 Pitch and sigma values for 130 nm features and 1:1 to 1:2.5 dutyratios. Duty Pitch (nm) axis sigma four-zone sigma 1:1 260 0.61 0.871:1.5 325 0.50 0.71 1:2 390 0.42 0.59 1:2.5 455 0.35 0.49

The masking aperture of this invention is a bilevel representation ofthe desired intensity distribution in the illuminator. It is desired toresemble a near continuously varying transition from open to opaqueareas. To achieve this result, the illumination pattern is divided intoa large number of elements and each element is a matrix of pixels.Dithering or pixelation of the continuous distribution of intensity isused for translation to the binary or bilevel masking aperture. Theelement array is large, consisting of, for instance, 5×5, 7×7, 9×9,11×11, 21×21 or 51×51 elements, but not limited to these cases. Theillumination profile is divided into such an element array. Individualmasking pixels are small, on the order of 10 to 100 μm, and are eithertranslucent or opaque. Their size is dependent on the size of thephysical masking aperture. The continuous tone nature of theillumination intensity profile is translated by controlling the spatialdensity of the bilevel display states on the masking aperture. Severaldecision rules may be implemented to produce the output distribution onthe masking aperture. A fixed threshold technique is simplest in form,but an ordered dithering approach may be used to most effectivelytranslate a continuous tone intensity profile into a bilevel maskingaperture representation. Intensity values are compared to aposition-dependent set of threshold values, contained in a n×n dithermatrix. A set of selection rules repeats the dither matrix in acheckerboard arrangement over the illumination field. One key to thisapproach is the generation of a bilevel representation of the continuoustone image with the minimal amount of low spatial frequency noise. Inother words, the occurrence of texture, granularity, or other artifactsis reduced to a minimum, allowing for the critical control ofillumination uniformity demanded in projection exposure tools.

The resulting bilevel representation of the continuous tone off-axisand/or on-axis illumination profile is then suitable for recording intoa photo-sensitive or electron beam-sensitive resist material through useof mask pattern generator. Other approaches might use lithographictechniques common to lithographic or printing technologies. Such aresist material, when coated over an opaque film on translucentsubstrate, can allow for pattern delineation and creation of the maskingaperture.

In the present invention, the existing intensity distribution at thepupil plane of an illumination system for a projection exposure tool ismodified through use of a bilevel masking aperture containing a maskingcell representation of a continuous tone intensity distribution. FIGS. 4and 5 show such distributions where four distributed-intensity zonesallow for off-axis and on-axis illumination of a photomask that containsgeometry oriented in horizontal and vertical directions only. A maximumcircular dimension is defined by a limiting zone, designed to limit themaximum off-axis angle projected onto the mask. This is used to balancethe off-axis illumination provided to the mask with the degree ofcoherency of the on-axis illumination. Smaller geometry requires higherlevels of on-axis partial coherence, leading to larger limiting zones.The extent of the zone will generally correspond to positions near orbeyond the maximum zone or angular position for the off-axisillumination, generally in, but not limited to, the range from 0.5 to1.0. The intensity at any position located beyond this limiting zone isset to zero. The shape of this limiting zone is not necessarilycircular, and selection of the shape will depend on the extent offeature orientation at the mask. Features constrained to one orientationonly require limitation of off-axis illumination in one direction,resulting in assigning a value of zero to any element beyond therequired x or y value corresponding to the limiting angle, as shown inFIG. 6. This allows for maximum energy at angles with the desired rangeand can lead to improved imaging and throughput performance. Featuresconstrained to two orientations only require that the aperture limitoff-axis illumination angles in two orthogonal directions, leading to anon-circular or square two dimensional character of the limiting zone asshown in FIG. 19. The invention allows for this tailoring of theillumination.

If the existing illumination intensity distribution at the pupil planeof the illumination system is not uniform, the non-uniformity at theplane can be deconvolved in accordance with the invention to result in amasking aperture that also incorporates compensation for non-uniformity.For example, many steppers provide a pupil that is guaranteed uniform(+/−1%) for only 80% of its full opening. At 85% open, the uniformity ofillumination may vary up to +/−10% or more. With the invention, thenon-uniformity may be canceled or reduced to an acceptable level.

The overlapping of the continuous intensity regions in the center of theillumination field produces the on-axis character required for lessdense features. The central intensity is generally greater than 0% andis commonly in the 10 to 50% range.

Illumination zones within the masking aperture control the illuminationto mask and are designed to produce optimal off-axis, on-axis, orcombined illumination. This invention allows for an infinite variety andnumber of such zones. Some are most desirable.

Zones may be circular, elliptical, 45 degree elliptical (that is,elliptical but oriented with axes at angles of 45 degrees and 135degrees), square, or other shapes dependent on the desired distributionof diffraction information to match mask geometry requirements orspecific lens behavior. The distribution of the energy in these zones orrings may be stepped, Gaussian, Lorentzian, or other similar shape. Thekurtosis of gaussian distributions may be normal (mesokurtic), narrow(leptokurtic), or flat-topped (platykurtic), or combinations of theseamong zones. Skewness, or departure from symmetry of the distributionmay be utilized for differential weighting of certain feature sizes.Circular symmetry may be best suited for most general cases andelliptical distributions can be utilized to accommodate x-ynonuniformities of the photomask or imparted by the projection lens (aresult for instance of astigmatic or comatic aberration). FIGS. 7 and 8show four elliptical normal distributed-intensity zones placed atdiagonal positions corresponding to off-axis illumination for geometryoriented in horizontal and vertical directions. FIG. 18 shows howrotated elliptical gaussian zones can offer improvements in intensitydistribution for x and y oriented geometry by concentrating some energyin the center of the pupil and the remaining energy distributed atfour-zone angles. Here, rotation of the zone axis allows for asymmetrical distribution. In this case, energy on the x/y axes islimited, reducing the non-optimal, on-axis illumination of targeteddense features. An increase in the efficiency of zero and firstdiffraction order overlap can result from such an illuminationdistribution.

For imaging of geometry in two directions only (x and y only forinstance), there is only a need to spread diffraction order informationin the direction of geometry. By limiting zone intensity distribution tox and y directions, resulting in continuous intensity or stepped-squareshaped zones, maximum off-axis illumination is maintained up to themaximum angle allowed by the zone dimensions. Beyond these angles, thedegree of off-axis illumination is limited and can be tailored morespecifically for the x and y oriented geometry. FIGS. 9 and 10 shows howstepped-square zones are implemented in an illumination profile. FIG. 11shows how this is translated into the bi-level representation.

A square or rectangular shaped obscuration (or an inner limiting zone)emphasizes the off-axis illumination for feature pitch values whosefrequency distribution falls beyond the chosen value (greater thanlambdu/(w*NA) where w is the full width obscuration value between 0 and2). This is shown in FIG. 21 for a Gaussian off axis distribution wherethe obscuration is 30% of the full aperture width. Combining a squareouter limiting zone and a square obscuration, an optimal condition ofoff axis illumination exists also. For features oriented in onedirection only, only two zones are needed on an axis opposite to thefeature direction. These zones can be slots or rectangles sincespreading of energy in the direction of feature orientation is of noconsequence to imaging performance and increases throughput. With twodimensional geometry, four slots are needed in x and y direction,resulting in a square ring, as shown in FIG. 22. This ring can also beconsidered as the combination of a square limiting zone and squareobscuration. This rectangular ring source distribution can deliveroff-axis illumination for features to 0.25 lambda/NA, depending on thechoice of the limiting outer square zone. This square ring sourcedistribution can also be combined with other off axis approaches, suchas a Gaussian our-zone design. FIG. 23 shows how a square ring sourcedistribution is added to a Gaussian four-zone design to produce resultsthat are common to both approaches (that is better performance for moredense features out to 0.25 lambda/NA and adequate through focus andthrough pitch imaging performance).

Other combinations of source distributions are possible. FIG. 24 showshow an annulur ring is combined with a gaussian four-zone design toemphasize the performance of more dense features than possible with theannular distribution alone. This approach can increase the off-axischaracter of the annulus and reduce the non-optimal on-axis character.

The masking aperture varies the intensity of the transmitted light atany element by modulating the state of pixels in each element. Thehighest intensity element has all pixels on or at maximum intensity.Light of suitable wavelength passes through without attenuation. In apreferred embodiment as shown in FIG. 12, an element with 64 pixels at aminimum intensity attenuates or blocks all light. Pixels of intensitybetween none and all are created by the number of pixels in a givenelement.

A masking aperture with a bilevel representation of such an illuminationdistribution is created by dithering the continuous tone images. Randomtechniques or fixed threshold techniques can be used. These fixedthreshold techniques are based on decision rules where any intensityvalue greater than a threshold value (T) results in a transparentmasking cell and a value less than T results in an opaque masking cell.The result is generally a high degree on banding in the bilevelrepresentation. A slight improvement over this method is to replace Twith equally distributed random numbers over the range 0 to S with a newrandom number generated for each intensity value. Less banding resultsbut signal to noise is low. A superior approach to the ditheredrepresentation is possible by comparing image intensity values toposition dependent thresholds contained in an n x n dither matrix,D^(n). For a D^(n) matrix, a matrix element D^(n) _(ij) is chosen basedon a rule set that causes the dither matrix to be repeated in acheckerboard fashion over the entire image with minimum low spatialfrequency noise. The proper choice of the dither matrix results inminimum texture or artifacts and maximum uniformity in intensity. Ingeneral, the optimum dither matrix is represented by the recursionrelationship: $D^{n} = {\begin{matrix}{{4D^{n/2}} + {D_{00}^{2}U^{n/2}4D^{n/2}} + {D_{01}^{2}U^{n/2}}} \\{{4D^{n/2}} + {D_{10}^{2}U^{n/2}4D^{n/2}} + {D_{11}^{2}U^{n/2}}}\end{matrix}}$

where: $U^{n} = {\begin{matrix}1 & 1 & \cdots & 1 \\1 & \quad & \quad & \quad \\\vdots & \quad & \quad & \quad \\1 & \quad & \quad & \quad\end{matrix}}$

To produce an 8×8 matrix to satisfy these optimization criteria, D8becomes: $D^{8} = {\begin{matrix}0 & 32 & 8 & 40 & 2 & 34 & 10 & 42 \\48 & 16 & 56 & 24 & 50 & 18 & 58 & 26 \\12 & 44 & 4 & 36 & 14 & 46 & 6 & 38 \\60 & 28 & 52 & 20 & 62 & 30 & 54 & 22 \\3 & 35 & 11 & 43 & 1 & 33 & 9 & 41 \\51 & 19 & 59 & 27 & 49 & 17 & 57 & 25 \\15 & 47 & 7 & 39 & 13 & 45 & 5 & 37 \\63 & 31 & 55 & 23 & 61 & 29 & 53 & 21\end{matrix}}$

To utilize this dithering matrix, for example, a continuous tone elementintensity distribution is divided into 64 levels. The lowest intensitylevel places a pixel in the masking element at the zero position. Anintensity value of 50% places pixels in the first 31 positions, and soforth. FIG. 12 schematically depicts the bilevel representation of 65gray levels using the ordered dithering algorithm. FIG. 26 shows howthis approach is used in a 51×51 element array. Other possibilities,such as 2-level, 4-level, 16-level, and so forth, are solved for in asimilar manner.

The performance improvement for 130 nm features using a 193 nmwavelength and 0.60 lens numerical aperture is described using anormalized aerial image log-slope NILS metric (normalized to the featuresize). This is the log of the slope of the intensity image (or aerialimage). FIGS. 13 and 14 show NILS plotted against focus position forcircular zone and a bi-level representation of distributed-intensityzone off-axis illumination where the central intensity is 26%. Theratios are the line to space size ratio, or duty ratio. The falloff ofNILS across all feature duty ratios matches more closely with increasingamounts of defocus for the distributed-intensity zone illuminationcompared to the circular zone condition. The resulting impact onlithographic imaging is a reduction in the dense to isolated lineproximity effect. FIG. 15a shows the imaging improvement achieved withthe square limiting zone using the illumination profile of FIG. 19. FIG.15b shows the further improvement using the hybrid design of FIG. 23.FIGS. 15a/d show a comparison of the square ring of FIG. 22 with anannular (circular) ring for a 248 nm wavelength, 0.63 NA & 150 nmfeatures at 1:1.5 duty ratio to 1:5 duty ratio. FIG. 15c shows theperformance for an annular or circular ring and FIG. 15d shows theperformance for the square ring. Illuminator dimensions are identical,only the shape differs. No circular or annular ring can match theperformance of the square ring.

The amount of total intensity allowed to pass through a masking aperturewill determine its acceptability in situations where exposure throughputis a concern, such as with the fabrication of integrated circuitdevices. Table 2 is a comparison of the throughput efficiency of severalvariations of the distributed-intensity four-zone approach, measuredrelative to conventional illumination with a δ of 0.7 and a strongcircular zone approach, where off-axis illumination is provided by fourcircular zones on a zero transmission field. The worst case throughputis for the circular zone four-zone design, where the total intensitythrough the pupil is 27% of that for conventional illumination withδ=0.7. The distributed-intensity four-zone approach with a 0.7 δcircular hard stop leads to 83% throughput and the same design with a0.7 half-width square hard stop results in 85% throughput. If the squarelimiting zone is increased in size to 0.8 half-width, the throughputincreases to 93% and imaging performance remains comparable to thecircular hard stop. This efficiency comes about because of the amount ofenergy allowed at the corners of the square pupil, where the diagonalapproaches the full extent of the condenser lens pupil, or a 6 valuenear 1.0. Comparison of intensity throughput is an important one asillumination modification is considered. If the illumination system ofan exposure tool can allow full value, δ=1 operation, this square hardstop variation of the distributed-intensity four-zone can lead tominimal losses.

TABLE 2 Relative Illumination approach Intensity Conventional, 0.7σ 1.00Circular four-zone 0.68σc, 0.2σ_(r), 0.8σ stop, 0% field 0.27 Circularfour-zone, 0.68σ_(c), 0.2σ_(r), 0.8σ stop, 25% field 0.52Distributed-intensity four-zone, 0.68σ_(c), 0.3σ_(r), 0.7σ circular 0.83stop Distributed-intensity four-zone, 0.68σ_(c), 0.3σ_(r), 0.7σ square0.85 stop Distributed-intensity four-zone, 0.68σ_(c), 0.3σ_(r), 0.8σsquare 0.93 stop

For on-axis conventional illumination, lens aberrations are evaluatedassuming full use of a lens pupil. With off-axis illumination,diffraction information is distributed selectively over the lens pupil,influencing the impact of aberrations on imaging. In general, astigmaticeffects can worsen while spherical aberration and defocus effects can beimproved. Coma induced image placement can be further aggravated withOAI unless rebalanced with tilt. FIG. 14 shows how the normalized imagelog slope is impacted for circular four-zone (δ_(c)=0.7 and δ_(r)=0.2 ona zero intensity field) with 0.1 waves of primary aberrations:astigmatism, tilt, spherical, and coma, through focus for a 130 nmfeatures imaged with a 193 nm imaging system, using a numerical apertureof 0.6. Although the effects of spherical aberration are minimal, theloss in NILS for astigmatism, tilt, and coma are significant. FIG. 17shows the same imaging process using the bi-level representation of thedistributed-intensity four-zone illumination described in thisinvention. Similar results are obtained using dithered intensitycircular, stepped square, and 45 degree elliptical zones. Here it can beseen that the influences of astigmatism and tilt are reduced at zerodefocus values and at larger defocus values (both positive and negative)the influences of all aberrations are reduced substantially, as comparedto the case shown in FIG. 16.

Implementation of the invention into existing illumination systems isaccomplished via access to the illumination optical system. In oneembodiment, pixilated half-tone illumination files are transferredlithographically onto a transparent substrate, such as fused silica,coated with a suitably opaque masking layer, such as chromium. Aphotoresist film coated over the metal coated transparent substrate isexposed using optical, electron beam, or other methods by translatingthe bi-level illumination representation into a suitablemachine-readable format. Photoresist development and subsequent etchingof the underlying masking film allows transfer of the pattern to themasking aperture. An anti-reflective layer can be coated over themasking film prior to photoresist coating, exposure, and processing toreduce reflection, stray light, and flare effects in the illuminationfield of the exposure tool. An anti-reflective layer can be coated overthe patterned aperture to match reflectances over the entireillumination field. Alignment of apertures is made possible byincorporating alignment fiducials on the masking apertures and on anaperture holders used to mount the invention into the exposure toolillumination system. Apertures can be inserted into as pupil plane ofthe illumination system.

Those skilled in the art will understand that a number of the newresults achieved with the above-described aperture mask may also beachieved with a traditional beamsplitter illumination system. Forexample, consider an illumination system such as the one shown anddescribed in U.S. Pat. No. 5,627,625. When an aperture mask 60 with alarge, central square opening 64 and an opaque border 62 is insertedproximate to the fly's eye or in the lens pupil 19, the four beams fromthe beam splitter 16 generate contour and three dimensional plotssimilar to FIGS. 18 and 19. The beams fill the corners of theillumination pupil and limit the non-optimal frequency spreadingcharacter along the x and y axes while optimizing the off-axisillumination angles. The new results can also be achieved with anillumination system using diffractive optical elements (DOEs) to shapethe illumination profile. An example of this illumination shaping methodis described, for instance, in U.S. Pat. No. 5,926,257 (Canon) and U.S.Pat. No. 5,631,721 (SVGL). FIG. 25 shows how an aperture (60) with alarge, central square opening (64) and an opaque border (62) is insertedproximate to the beam shaping optical system (2). This can allow forshaping with a square limiting zone. Additionally, the beam shapingoptical system using diffractive optical elements can be tailored toproduce similar results. That is especially helpful for imaging featuresthat are oriented along x and y directions in the mask plane. The use ofa central obscuration (square and also round shaped) can similarly beachieved and can lead to performance improvements described here.Furthermore, any combination of off-axis illumination with a squareaperture or obscuration has potential to improve performance forgeometry oriented in the x-y direction. This can include, but is notlimited to, round zones, elliptical zones, square zones, and annularslots (that is an annular ring masked off on x and y axis to formarc-shaped zones).

The masking aperture described above may be used alone or in combinationwith prior art techniques and apparatus. For example, the maskingaperture of the invention may be combined with the four hole metal maskdescribed in JP patent Laid-Open (KOKAI) Publication No. 4-267515 anddiscussed in the background of U.S. Pat. No. 5,627,625.

Those skilled in the art will understand that the new results achievedwith elliptical, 45 degree elliptical, square rings, and square shapedzones may also be achieved with alternative approaches, including thebeamsplitter approach. This could be accomplished for example with theillumination system shown in FIG. 20 (and described in U.S. Pat. No.5,627,625) by shaping divided beams into elliptical, 45 degreeelliptical, and square shapes in the beamsplitter unit 16 andsuperposing them using a prism unit 17. Gaussian or similar shapedenergy distribution is possible.

Those skilled in the art will also understand that the new resultsachieved with elliptical, 45 degree elliptical, square rings, and squareshaped zone may also be achieved with diffractive optical elementapproaches to beam shaping, such as that described in U.S. Pat. No.5,926,257 and U.S. Pat. No. 5,631,721 by tailoring the diffractiveoptical elements to exhibit these characteristics. The micro-diffractiveoptical elements within the beam-shaping optical system (2) in FIG. 25are manipulated to allow for the required shaping. It is well known thatdiffractive optical elements (DOEs) and holographic optical elements(HOEs) can allow for efficient manipulation of arbitrary wavefronts withmore flexibility and reduced fabrication requirements compared toconventional refractive optics (see for instance Z. Yang and K.Rosenbruch, SPIE Vol. 1354, (1990), 323). One DOE or HOE or thecombination of two or more DOEs or HOEs allow for the design flexibilityneeded to achieve the desired results, as demonstrated for instance inU.S. Pat. No. 5,926,257 and in SPIE Vol. 1354, (1990), 323.

The present invention is described above but it is to be understood thatit is not limited to these descriptive examples. The numerical values,number of zones, shapes, and limiting zones may be changed toaccommodate specific conditions of masking, aberration, featureorientation, duty ratio requirements, lens parameters, initialillumination non-uniformities, and the like as required to achieve highintegrated circuit pattern resolution. Results can also be obtained bycontrolling illumination at any Fourier Transform plan in theillumination system.

What is claimed:
 1. An illumination system comprising: a light source;and a beam shaping optical system arranged in an optical path from saidlight source; wherein said beam shaping optical system comprises amasking aperture comprising: a translucent substrate; a half-tonedithered pattern on the substrate, said half-tone dithered patterncomprising an array of pixels, each pixel of a clear or opaque type andof the same size, said clear and opaque pixels for respectively passingand blocking incident light, wherein the number, size, and type of thepixels are chosen in accordance with: (a) the wavelength of light usedto illuminate the photomask, and (b) the size and shape of the featuresof the photomask.
 2. The illumination system of claim 1, wherein thebeam shaping optical system comprises a plurality of masking apertures.3. The illumination system of claim 2, wherein at least one of themasking apertures comprises an opaque plate with one or more apertures.4. The illumination system of claim 3, wherein at least one of themasking apertures comprises a translucent plate with a centralobscuration.
 5. The illumination system of claim 4, wherein the centralobscuration is circular or square.
 6. The illumination system of claim1, further comprising an optical integrator arranged on a light exitside of said beam shaping optical system.
 7. The illumination system ofclaim 6, further comprising a square shaped aperture arranged betweensaid beam shaping optical system and said optical integrator.
 8. Theillumination system of claim 6, wherein the optical integrator is afly's eye array of lenslets.
 9. The illumination system of claim 1,wherein the beam shaping optical system is constructed to produce ashaped illumination pattern that has a shape selected from the groupconsisting of round, square, and elliptical shapes.
 10. The illuminationsystem of claim 9, wherein the beam shaping optical system comprises adiffractive optical element.
 11. The illumination system of claim 9,wherein the beam shaping optical system comprises a beam splitterlocated between the source of light and an exit pupil of saidillumination system.
 12. The illumination system of claim 1, furthercomprising a relay optical system arranged between said light source andan exit pupil of said illumination system.
 13. The illumination systemof claim 1, wherein the half-tone dithered pattern comprises an array ofdiffraction elements and each diffraction element is a dithered patternof clear or opaque pixels.
 14. The illumination system of claim 13,wherein each diffraction element pixel comprises an n×n dithered matrixof pixels, the intensity of each element is defined by the number andtype of pixels in its dithered matrix and wherein the pixels in eachmatrix are dithered to avoid artifacts.
 15. The illumination system ofclaim 13, wherein the relative intensity of each subpixel is defined bya recursion relationship where: ${D^{n} = {{\begin{matrix}{{4D^{n/2}} + {D_{00}^{2}U^{n/2}4D^{n/2}} + {D_{01}^{2}U^{n/2}}} \\{{4D^{n/2}} + {D_{10}^{2}U^{n/2}4D^{n/2}} + {D_{11}^{2}U^{n/2}}}\end{matrix}}\quad {and}}}\quad$ $U^{n} = {{\begin{matrix}1 & 1 & \cdots & 1 \\1 & \quad & \quad & \quad \\\vdots & \quad & \quad & \quad \\1 & \quad & \quad & \quad\end{matrix}}.}$


16. The illumination system of claim 15, wherein the matrix of pixelscomprises an 8×8 matrix and the relative intensity, D⁸, comprises:$D^{8} = {{\begin{matrix}0 & 32 & 8 & 40 & 2 & 34 & 10 & 42 \\48 & 16 & 56 & 24 & 50 & 18 & 58 & 26 \\12 & 44 & 4 & 36 & 14 & 46 & 6 & 38 \\60 & 28 & 52 & 20 & 62 & 30 & 54 & 22 \\3 & 35 & 11 & 43 & 1 & 33 & 9 & 41 \\51 & 19 & 59 & 27 & 49 & 17 & 57 & 25 \\15 & 47 & 7 & 39 & 13 & 45 & 5 & 37 \\63 & 31 & 55 & 23 & 61 & 29 & 53 & 21\end{matrix}}.}$


17. The illumination system of claim 1, further comprising an opticalintegrator arranged in an optical path from said beam shaping opticalsystem; and a square-shaped aperture disposed between said opticalintegrator and said beam shaping optical system.
 18. A lithographicapparatus comprising, an illumination system arranged to illuminate amask; and a projection optical system arranged to project radiation fromsaid mask onto a substrate, wherein said illumination system comprises abeam shaping optical system having a masking aperture comprising: atranslucent substrate, and a half-tone dithered pattern on thesubstrate, wherein said half-tone dithered pattern on said substratecomprises an array of pixels, each pixel being a clear or opaque typeand of the same size, said clear and opaque pixels being for passing andblocking respectively incident light, wherein the number, size, and typeof the pixels are chosen in accordance with: (a) the wavelength of lightused to illuminate the photomask, and (b) the size and shape of thefeatures of the photomask.
 19. The lithographic apparatus of claim 18,wherein the beam shaping optical system comprises a plurality of maskingapertures.
 20. The lithographic apparatus of claim 19, wherein at leastone of the masking apertures comprises an opaque plate with one or moreapertures.
 21. The lithographic apparatus of claim 20, wherein at leastone of the masking apertures comprises a translucent plate with acentral obscuration.
 22. The lithographic apparatus of claim 21, whereinthe central obscuration is circular or square.
 23. The lithographicapparatus of claim 18, wherein said illumination system furthercomprises an optical integrator arranged on a light exit side of saidbeam shaping optical system.
 24. The lithographic apparatus of claim 23,wherein said illumination system further comprises a square shapedaperture arranged between said beam shaping optical system and saidoptical integrator.
 25. The lithographic apparatus of claim 23, whereinthe optical integrator is a fly's eye array of lenslets.
 26. Thelithographic apparatus of claim 18, wherein the beam shaping opticalsystem is constructed to produce a shaped illumination pattern that hasa shape selected from the group consisting of round, square, andelliptical shapes.
 27. The lithographic apparatus of claim 26, whereinthe beam shaping optical system comprises a diffractive optical element.28. The lithographic apparatus of claim 26, wherein the beam shapingoptical system comprises a beam splitter located between the source oflight and an exit pupil of said illumination system.
 29. A lithographicapparatus, of claim 18, further comprising a relay optical systemarranged between said light source and an exit pupil of saidillumination system.
 30. The lithographic apparatus of claim 18, whereinthe half-tone dithered pattern comprises an array of diffractionelements and each diffraction element is a dithered pattern of clear oropaque pixels.
 31. The lithographic apparatus of claim 18, wherein saidillumination system further comprises: an optical integrator arranged inan optical path from said beam shaping optical system; and asquare-shaped aperture disposed between said optical integrator and saidbeam shaping optical system.
 32. The lithographic apparatus of claim 31,wherein each diffraction element pixel comprises an n×n dithered matrixof pixels, the intensity of each element is defined by the number andtype of pixels in its dithered matrix and wherein the pixels in eachmatrix are dithered to avoid artifacts.
 33. The lithographic apparatusof claim 31, wherein the relative intensity of each subpixel is definedby a recursion relationship where: ${D^{n} = {{\begin{matrix}{{4D^{n/2}} + {D_{00}^{2}U^{n/2}4D^{n/2}} + {D_{01}^{2}U^{n/2}}} \\{{4D^{n/2}} + {D_{10}^{2}U^{n/2}4D^{n/2}} + {D_{11}^{2}U^{n/2}}}\end{matrix}}\quad {and}}}\quad$ $U^{n} = {{\begin{matrix}1 & 1 & \cdots & 1 \\1 & \quad & \quad & \quad \\\vdots & \quad & \quad & \quad \\1 & \quad & \quad & \quad\end{matrix}}.}$


34. The lithographic apparatus of claim 30, wherein the matrix of pixelscomprises an 8×8 matrix and the relative intensity, D ⁸ comprises:$D^{8} = {{\begin{matrix}0 & 32 & 8 & 40 & 2 & 34 & 10 & 42 \\48 & 16 & 56 & 24 & 50 & 18 & 58 & 26 \\12 & 44 & 4 & 36 & 14 & 46 & 6 & 38 \\60 & 28 & 52 & 20 & 62 & 30 & 54 & 22 \\3 & 35 & 11 & 43 & 1 & 33 & 9 & 41 \\51 & 19 & 59 & 27 & 49 & 17 & 57 & 25 \\15 & 47 & 7 & 39 & 13 & 45 & 5 & 37 \\63 & 31 & 55 & 23 & 61 & 29 & 53 & 21\end{matrix}}.}$